The astrophysical reaction 8 Li(n,γ) 9 Li from measurements by reverse kinematics

Transcription

1 J. Phys. G: Nucl. Part. Phys. 25 (1999) Printed in the UK PII: S (99) The astrophysical reaction 8 Li(n,γ) 9 Li from measurements by reverse kinematics Carlos A Bertulani Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil bertu@if.ufrj.br Received 22 December 1998, in final form 17 June 1999 Abstract. We study the breakup of 9 Li projectiles in high-energy (28.5 MeV/u) collisions with heavy nuclear targets ( 208 Pb). The wavefunctions are calculated using a simple potential model for 9 Li. A good agreement with the measured data is obtained with insignificant E2 contribution. The reaction 8 Li(n,γ) 9 Li competes with the 8 Li(α, n) 11 B reaction in the inhomogeneous Big Bang model, which assumes the existence of proton-rich and neutron-rich regions of space during the early Big Bang. This reaction also competes with the 8 Li β-decay in the r-process. Depending on the rate, this reaction may affect the primordial abundance and reduce the stellar production of A>12 nuclei by as much as 50% [1]. The Coulomb dissociation method [2,3] has proven to be a useful tool for extracting radiative capture reaction cross sections of relevance in nuclear astrophysics. In particular, it appears that the Coulomb dissociation of 9 Li is very useful [4] for elucidating the role of the inhomogeneous nucleosynthesis in the Big Bang model the formation of 9 Li via the 8 Li(n,γ) 9 Li reaction. However, a few lingering questions still need to be addressed, including the importance of E2 excitations for the MSU experiment [4], performed at approximately 28.5 MeV/u. We attempt to resolve this issue by using a relatively simple but still realistic nuclear model that, however, yields agreement with data and suggests that E2 excitations are negligible for the kinematical conditions of the MSU experiment. For 9 Li we adopt a single-particle model in which the J 0 = 3/2 ground state can be described as a j 0 = p 3/2 neutron interacting with the 8 Li core. The core is assumed to be inert with an intrinsic spin I c = 2 +. A realistic value for the spectroscopic factor for this configuration is about 0.94 [5]. Since we are only interested in a rough estimate of the E2 contribution in the Coulomb breakup of 9 Li, we take S = 1. The single particle states, R Ex lj (r), for the excitation energy E x = E n + E 0, where E n is the neutron 8 Li relative energy, are found by solving the Schrödinger equation with a nuclear + spin orbit potential, given by [ V(r)=V 0 1 F s.o. (l s) r 0 d r dr ] f (r), f (r) = [ 1+exp ( )] r R 1, (1) with parameters a = 0.52 fm and r 0 = 1.25 fm. The radius of the 8 Li core, R = fm, was chosen so that the radiative capture cross section would fall in the range of the experiment of [4]. The spin orbit strength is set to F s.o. = fm. This is consistent with the choice /99/ $ IOP Publishing Ltd 1959 a

2 1960 C A Bertulani for the 8 B model used in [6] but is slightly smaller than the value 0.38 fm which works well for the low-lying spectra of 17 O, 13 C, and 11 Be [7]. The well depth for the ground state l 0 (j 0 I c )J 0 = (p 3/2, 2 + )3/2, was adjusted to reproduce the one-neutron separation energy, E 0 = 4.05 MeV, and it is V 0 ((p 3/2, 2 + )3/2 ) = MeV. A neutron-unbound state exists in 9 Li at 4.31 MeV [8, 9] above the neutron decay threshold. Its predicted spin parity is 5/2. These are described as p 3/2 waves coupled to the ground state of the core, and the well depth for this channel, V 0 = MeV has been adjusted to reproduce the resonance energy and its predicted [9] width, Ɣ = MeV. For all other partial waves (s 1/2, p 1/2, d 3/2, etc) we choose identical well depths and set them equal to the value MeV, as for the ground state. To compute the S-factors for the capture process b + x a we have used the first-order perturbation theory. The matrix elements for electric multipole states (we couple angular momentum as l + s = j, and j + I c = J) are given by JM M(Eλµ) J 0 M 0 = e λ ( 1) j 0+I c +J 0 +l 0 +l+λˆλˆ j 0 Jˆ 0 4π ĵ ( (J 0 M 0 λµ JM) j 0 λ 1 ){ } j J 2 0 j1 Ic O(1 2;λ) (2) 2 J 0 j 0 λ where O(1 2; λ) = φlj J (r)φj 0 E 0 l 0 j 0 (r)r λ dr is the overlap integral, ˆm 2m + 1, and e λ = Z b e( A x /A a ) λ + Z x e(a b /A a ) λ is the effective electric charge (b 8 Li, x n). Here R Ex lj (r) is the radial wavefunction for the relative motion of the neutron and the core, normalized to 2m bx /π h 2 k sin(kr + δ lj ) at large r, where k is the relative momentum. The response functions (multipole strengths) for the excitation of 9 Li are obtained by summing over all partial waves db(eλ) = de x lj where db(eλ;l 0 j 0 E x lj ) = m bx de x h 2 k db(eλ;l 0 j 0 E x lj ) de x, (3) (2J +1) J { } j J Ic j M(Eλ) j J 0 j 0 λ 0 2 J. (4) The sum over J in the last equality reduces to j M(Eλ) j 0 2 /(2j 0 +1)if the single-particle matrix elements are independent of the channel spin J. The multipole strengths are presented in figure 1 for E1 and E2 excitations, as functions of the neutron energy, E n = h 2 k 2 /2m bx. They are dominated by s-wave components: the higher angular momentum waves become relevant for higher energies, as expected. This result has also been obtained in previous calculations [8, 10]. The quadrupole strength shows a resonance peak located at the excitation energy, E x = MeV. It is divided by ten in order to be displayed in the same figure. The cross sections for direct capture are given by σ (λ) DC (E x) = (2I c +1) (2π) 3 ( ) (λ +1) 1 2λ+1 Ex db(eλ). (5) (2J 0 +1) λ[(2λ+1)!!] 2 k 2 hc de x Figure 2 displays the direct capture cross section for the E1 transitions. The experimental data are from [4]. Due to the factor (E x / hc) 2λ+1 appearing in equation (5) (and since the peak of db(e2) fm 4 is one order of magnitude greater than db(e1) fm 2, as seen from figure 1), the de E2 contribution to the radiative capture cross section is about 3 4 orders of magnitude smaller than the E1 contribution. de

3 The astrophysical reaction 8 Li(n, γ ) 9 Li 1961 Figure 1. Response function (equation (3)) in units of e 2 fm 2 MeV 1 (e 2 fm 4 MeV 1 ) for E1(E2) transitions in the reaction 9 Li(γ, n) 8 Li, as a function of the neutron energy relative to the 8 Li core. The lower curves are the f (solid) and p + f (dashed) waves contribution, while the upper curves display the contribution of s (solid) and s+d (dashed) waves. Figure 2. Radiative capture cross sections, in µb, for the reaction 8 Li(n,γ) 9 Li in the direct capture model. The solid curve is obtained with s-waves, while the dashed curve includes d-wave transitions. Since there are no data for the elastic scattering of 9 Li on Pb targets at this bombarding energy, we construct an optical potential using an effective interaction of the M3Y type [12,13] modified so as to reproduce the energy dependence of total reaction cross sections, i.e. [13], t(e,s) = i hv σ NN (E)[1 iα(e)]t(s), (6) 2t 0 where t 0 = 421 MeV fm 3 is the volume integral of the M3Y interaction t(s), s is the nucleon nucleon separation distance, v is the projectile velocity, σ NN is the nucleon nucleon cross section, and α is the real-to-imaginary ratio of the forward nucleon nucleon scattering amplitude. At 28.5 MeV/nucleon, we use σ NN = 20 fm 2 and α = The optical potential is given by U(E,R) = d 3 r 1 d 3 r 2 ρ P (r 1 )ρ T (r 2 )t(e, s), (7) where s = R + r 2 r 1, and ρ T (ρ P ) is the ground state density of the target (projectile). Following [14], the Coulomb amplitude is given by f C = (J M) f λµ, (8) λµ

4 1962 C A Bertulani where (J M) f λµ = i 1+µ Z ( ) T em λ PT Ex 2λ +1exp{ iµφ} µ h 2 (q) hc ( c ) G Eλµ JM M Eλ, µ J 0 M 0, (9) v ( ) Ex b µ (q) = dbbj µ (qb)k µ exp(iχ(b)), (10) 0 γ hv q = 2k sin(θ/2), θ (φ) is the (azimuthal) scattering angle, and m PT is the reduced mass of the target + projectile. J µ (K µ ) is the cylindrical (modified) Bessel function of order µ, and the functions G πλµ (c/v) are tabulated in [15]. The angular momentum algebra connects JM M Eλ, µ J 0 M 0 with the reduced matrix elements of equation (2). The eikonal phase, χ(b),isgivenby χ(b) = 2ηln(kb) 1 dzu opt (R), (11) hv where η = Z P Z T e 2 / hv, hk is the projectile momentum, and R = b 2 + z 2. The optical potential, U opt, in the above equation is given by equation (7). The cross section for Coulomb excitation of a state with angular momentum J and excitation energy E x is obtained by an average (and a sum) over the initial (final) angular momentum projections: d 2 σ Eλ 1 = d de x 2J 0 +1 M 0,M (J M) f λµ 2. (12) As explained in detail in [2, 3, 14], the above cross section can be factorized in terms of a product of virtual photon numbers and breakup cross sections by real photons, which by detailed balance is directly related to the radiative capture cross sections. Thus a measurement of dσ Eλ /d de x can be used to obtain radiative capture cross sections of astrophysical interest. Experimentally, the nuclear contribution to the breakup cross section can be separated by repeating the measurement on light targets (see, e.g., [4]). At the bombarding energies of tens of MeV/nucleon, the E2 virtual photon number is much larger than that of E1. As a consequence, even when the E2 contribution to the radiative capture cross section is small, it may be amplified in Coulomb breakup experiments [3]. A known example is the Coulomb breakup of 8 B [16] at 50 MeV/nucleon, which has been used to extract the S-factor for the radiative capture cross section 7 Be(p,γ) 8 B at low energies. It has been claimed [17] that the E2 contribution accounts for as much as 20% of the differential cross sections for the experimental conditions. This poses an additional experimental problem, since one needs to separate the wanted E1 transition matrix elements from the unwanted E2 transition matrix elements. Nonetheless, it was shown in [18] that the E2 contribution in the Coulomb breakup of 8 B is much less than that predicted in [17]. In order to infer the relevance of E2 for the breakup of 9 Li (28.5 MeV) on lead targets, we plot in figure 3 the angle integrated cross section dσ λ /de x, as a function of the neutron energy relative to 9 Li. The quadrupole contribution has been multiplied by ten in order to be more visible. We see that the E2 contribution to the breakup cross section is at least one order of magnitude smaller than the E1 contribution, even at the resonance region. We thus conclude that E2 transitions are not relevant in the experiment of [4]. Although we obtained this result by means of a single-particle model for the 8 Li(n,γ) 9 Li reaction, we do not expect that it would change appreciably with more sophisticated models.

5 The astrophysical reaction 8 Li(n, γ ) 9 Li 1963 Figure 3. Coulomb breakup cross section dσ Eλ /de (in mb MeV 1 ) for the reaction 9 Li (28.5 MeV/nucleon) + Pb 8 Li + n + Pb, as a function of the neutron 9 Li relative energy, in MeV. The E2 breakup contribution multiplied by ten is also shown. Acknowledgments I am grateful to Aaron Galonsky for helpful comments and suggestions. This work was supported in part by MCT/FINEP/CNPQ(PRONEX) under contract No , and by the Brazilian funding agencies FAPERJ and FUJB. References [1] Malaney R A and Fowler W F 1988 The Origin and Distribution of the Elements ed G J Mathews (Singapore: Word Scientific) [2] Baur G, Bertulani C A and Rebel H 1986 Nucl. Phys. A [3] Bertulani C A and Baur G 1988 Phys. Rep [4] Zecher P D et al 1998 Phys. Rev. C [5] Brown B A Private communication [6] Bertulani C A 1996 Z. Phys. A [7] Esbensen H, Brown B A and Sagawa H 1995 Phys. Rev. C [8] Mao Z Q and Champagne A E 1991 Nucl. Phys. A [9] Kumar N 1983 Nucl. Phys. A [10] Descouvement P 1993 Astrophys. J. Lett [11] Koester L and Steyerl A 1977 Neutron Physics (Berlin: Springer) [12] Kobos A M, Brown B A, Lindsay R and Satchler G R 1984 Nucl. Phys. A [13] Bertulani C A, Lotti P and Sagawa H 1998 Phys. Rev. C [14] Bertulani C A and Nathan A M 1993 Nucl. Phys. A [15] Winther A and Alder K 1979 Nucl. Phys. A [16] Motobayashi T et al 1994 Phys. Rev. Lett [17] Langanke K and Shoppa T D 1994 Phys. Rev. 49 R1771 [18] Gai M and Bertulani C A 1995 Phys. Rev. C